Evaluating Int Cot X Csc 2x Mathrm D X With U Cot X

Is there a closed-form expression for the series $$ \\sum_{k=0}^n\\binom\\alpha k^2\\lambda^k,\\quad \\alpha ~ \\text{is non-integer} $$ There is an identity involving binomial …. Insights reveal, $$ I_n=\int_ {1}^ {\infty}\frac {n\bigl (x^ {\alpha+1}-x^\alpha\bigr)\sin\!\left (\frac {1} {x}-1\right)} {x^3\bigl (x^\alpha+n^\alpha (x-1)^\alpha\bigr)}\,dx, \qquad. Observations indicate, How would I go about evaluating this integral? $$\int_0^ {\infty}\frac {\ln (x^2+1)} {x^2+1}dx. Additionally, How would you evaluate the following series? $$\\lim_{n\\to\\infty} \\sum_{k=1}^{n^2} \\frac{n}{n^2+k^2} $$ Thanks. These findings regarding Evaluating Int Cot X Csc 2x Mathrm D X With U Cot X provide comprehensive context for understanding this subject.